#1082697

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(2,3)(5,6,9,8)(7,11,12,10)(13,14,18,17)(15,20,16,19);(1,2,4,3)(5,6,9,8)(7,10,12,11)(15,16)(19,20)(22,25)(26,28);(2,3)(13,15,18,16)(14,19,17,20)(21,22,24,25)(23,26,27,28);(5,7,9,12)(6,10,8,11)(13,16,18,15)(14,20,17,19);(1,3,4,2)(5,8,9,6)(7,11,12,10)(13,17,18,14)(15,20,16,19)(21,23,24,27)(22,26,25,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fh,d^2=i,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 989893);